Table 3 Scenarios for Example 2

Stratification/Proxy Z (with J strata)

Environmental exposure E and Gene proxy S_G

J = 4

Z = 1: E^- S_G ^-, Z = 2: E^- S_G ⁺, Z = 3: E⁺ S_G ^-, Z = 4: E⁺ S_G ⁺

Phase One prevalence among controls (τ⁰ _j):

τ⁰ ₁ = Pr(E^-)Pr(S_G ^-) = (1 - P_E)[(1-Se)P_G+Sp(1-P_G)]

P_E = 20%

τ⁰ ₂ = Pr(E^-)Pr(S_G ⁺) = (1 - P_E)[SeP_G+(1-Sp).(1-P_G)]

P_G = 1%

τ⁰ ₃ = Pr(E⁺)Pr(S_G ^-) = P_E[(1-Se)P_G+Sp(1-P_G)]

τ⁰ ₄ = Pr(E⁺)Pr(S_G ⁺) = P_E[SeP_G+(1-Sp).(1-P_G)]

Risk factor X (with K outcomes)

Exposure to E and exposure to G: K = 4

X = 1: E^- G^-, X = 2: E^- G⁺, X = 3: E⁺ G^-, X = 4: E⁺ G⁺

Disease Model (Odds Ratios ψ_k)

ψ₁ = 1, ψ₂ = 3, ψ₃ = 2, ψ₄ = ψ₂ × ψ₃ × OR_I = 30

Phase Two prevalence of X among controls by stratum (π⁰ _jk)

Z = 1: π⁰ ₁₁ = (1 -P_E)Sp(1-P_G)/Pr(S_G ^-),

π⁰ ₁₂ = 1 - π⁰ ₁₁, π⁰ ₁₃ = π⁰ ₁₄ = 0

Z = 2: π⁰ ₂₁ = (1 - P_E)(1 - Sp)(1-P_G)/Pr(S_G ⁺),

π⁰ ₂₂ = 1 - π⁰ ₂₁, π⁰ ₂₃ = π⁰ ₂₄ = 0

Z = 3: π⁰ ₃₁ = π⁰ ₃₂ = 0, π⁰ ₃₃ = P_E Sp(1-P_G)/Pr(S_G ^-),

π⁰ ₃₄ = 1 - π⁰ ₃₃

Z = 4: π⁰ ₄₁ = = π⁰ ₄₂ = 0, π⁰ ₄₃ = P_E (1 - Sp)(1-P_G)/Pr(S_G ⁺),

π⁰ ₄₄ = 1 - π⁰ ₄₃